q-series, elliptic curves, and odd values of the partition function

摘要

Letp(n)be the number of partitions of an integern. Euler proved the following recurrence forp(n):p(n)=∑k=1∞(−1)k+1(p(n−ω(k))+p(n−ω(−k))),                                        (*)whereω(k)=(3k 2+k)/2. In view of Euler's result, one sees that it is fairly easy to computep(n)very quickly. However, many questions remain open even regarding the parity ofp(n). In this paper, we use various facts about elliptic curves andq-series to construct, for everyi≥1, finite setsMifor whichp(n)is odd for an odd number ofn∈Mi.